# A theoretical and empirical analysis of family migration and household production: U.S. 1980-1985.

I. Introduction

Migration studies have usually focussed on individual persons or workers. There has, however, been some interest in family migration. While single individuals in the labor market only have to worry about their own income, families, where both the husband and wife may be in the labor market, need to be concerned with the income of both spouses. Hence, double income families may be less likely to move than single income families. Jacob Mincer |26~ develops and explores this hypothesis, to be called the tied-mover hypothesis.(1) The tied-mover hypothesis has recently been extended to include not only considerations of the wife's earnings but also considerations of her household production as factors in the family's migration decisions |38~. In this paper, the research on family migration will be extended in three ways. A formal model of household production and family migration will be developed and used to generate testable hypotheses about family migration. Variables, which otherwise would not have a clear economic interpretation, are incorporated into the model. The model will then be estimated and some policy implications will be discussed. One policy implication concerns the efficiency of the labor market. Labor market efficiency needs to be judged not only in terms of the efficiency of the job search of single individuals but also in terms of the efficiency of the joint job search of double income families.

This paper is organized into five sections. First, a comparative static household production model of locational choice is developed.(2) Next, this model of locational choice is used to develop a theoretical model of family migration. Third, the data and empirical model are discussed. Fourth, the empirical results are reported and interpreted in terms of three different types of migration. Estimates of the impact of selected variables on the distance of the move are also considered. The final section summarizes the general findings and offers some policy implications.

II. A Model of Locational Choice

In this section, a comparative static model of family migration is developed using a Becker-Lancaster household production approach |2; 18~. The family will be assumed to live at its optimum location. The technology of the household and market variables, such as prices, the husband's income and the wife's wage rate, will be assumed to depend upon the family's location. The family moves when these location-specific variables change in a way which makes the current location no longer optimal. We will see, in section V, that this approach is capable of explaining the patterns of family migration which are observed.

In a standard household production model, the household receives utility directly from the commodities which it consumes. Commodities are produced from inputs which consist of goods purchased in the market and the household's own time. A household maximizes its utility from consuming commodities subject to a linear income constraint and the household's technology. This model can be applied to migration by introducing, into the household's production functions, characteristics of family members and of the location of the household. Consider an example where a household, with both a husband and a wife, maximizes

U = U(|Z.sub.1~,|Z.sub.2~,...|Z.sub.n~) (1a)

subject to

B = Y + w(T - |summation of~ (|t.sub.i~) where i = 1 to n) = |summation of~ (|p.sub.j~|X.sub.j~) where j = 1 to m + hH, (1b)

and

|Z.sub.i~ = |f.sub.i~(|X.sub.1i~,|X.sub.2i~,...|X.sub.mi~;H;|t.sub.i~/|Thet a~,L), for i = 1,2,...n, (1c)

where

|X.sub.j~ = |summation of~ (|X.sub.ji~) where i = 1 to n, for j = 1,2,...m. (1d)

U is family utility, |Z.sub.i~ is the ith commodity, |f.sub.i~ is the household production function for the ith commodity, |X.sub.ji~ is the quantity of good j used to produce commodity i, |p.sub.j~ is the price of good j, |t.sub.i~ is the wife's time used to produce commodity i, w is the market wage rate of the wife, T is the total time the wife has at her disposal, Y is the husband's income which includes non-wage family income, B is the family budget, h is the price of housing services, H is housing services, |Theta~ is a vector of household characteristics, and L is a vector of relevant attributes of the family's current location.(3) Note that some household characteristics such as education will influence both the household production function, and the earnings variables, w and Y.(4)

Many of the exogenous variables in the above constraints depend upon the family's location and will change if the family moves to another location. Location will influence family utility in two ways. First, market prices, housing costs, the husband's income and the wife's wage rate depend upon location. Second, location will have a direct effect on household production possibilities because elements in L, which can differ from one location to the next, are arguments in the production functions of the household. Included in L are such factors as climate, proximity to friends and relatives, and the availability of cultural, recreational and educational amenities. Note that the impact that location has on Y, w, and production possibilities will be affected by household characteristics, |Theta~, such as marital status, the number of children, the ages of family members, and the education of family members.

If the production functions are linearly homogeneous and the wife does not completely specialize in household production, then the optimal production of commodities implies that the constraints in (1) can be combined into a single, linear constraint,

I = |Pi~Z, (2)

where I = Y + wT is called full income, |Pi~ is a vector of implicit (shadow) prices of commodities and Z is a vector of commodities.(5) The shadow prices depend upon Y, w, h, |Theta~, L, and a vector of market prices, p.

In comparing locations, families, which are living at their optimal location, receive rent in the sense that there is some amount of money which would be just sufficient to induce them to move |40~.(6) This implicit rent will be called locational rent. To more concretely define locational rent, suppose there are only two locations. Let I and I|prime~ be full income at each location, and let |Pi~ and |Pi~|prime~ be scalars representing the cost of commodity production at each location. Since locational rent would equate real commodity consumption at both locations, locational rent can be written as

R = I(|Pi~|prime~/|Pi~) - I|prime~, (3)

where R is locational rent.

To illustrate the role locational rent plays in locational choice, consider an example where there are two periods of time. At the beginning of the first period, the family is assumed to be living at its optimal location. Hence, initial locational rent is positive, given the vector of exogenous variables governing the family's choice for that period. At the end of the period, the family will face a new vector of exogenous variables and, hence, would receive a new level of locational rent. If this new level of locational rent is negative, the family moves.

A basic assumption of the migration model developed here is that the larger the locational rent, the less likely any given change in the family's economic situation will be of sufficient magnitude to cause a move. For example, an increase in the income that the husband could earn at another location will reduce locational rent but will not necessarily cause the family to move. The increase in income must exceed the original level of locational rent if the family is to move.(7)

III. The Migration Model

Since household migration is assumed to be a function of locational rent, where locational rent is a function of market variables, household characteristics, and locational characteristics, the impact that these variables have on migration can be discussed in terms of how they affect locational rent. The basic underlying model is

M = f(R), (4)

where M is the probability that the family will move by the end of a given period and R is locational rent at the beginning of the period.

It is by no means straight forward to estimate equation (4). Three difficulties arise. First, there are more than two locations. Second, the market variables for the family at an alternative location, denoted as Y|prime~, p|prime~, and w|prime~, are not known to the investigator. Third, the model is intrinsically nonlinear. The model does, however, have some clear implications as to the direction of the impact which the known variables have on migration.

The direction of this impact can be found by differentiating R with respect to the relevant variables. To do so, consider a simple example where there is only one aggregate commodity produced by a Cobb-Douglas production function

Z = A|H.sup.|Alpha~~|X.sup.|Beta~~|t.sup.1-|Alpha~-|Beta~~ (5)

with inputs of market goods, the wife's time, and housing.(8) Furthermore, assume there are only two locations; the optimal current location and a suboptimal alternative location. Finally, to simplify the discussion of regional differences in household technology, assume that the only difference in these technologies is the value of A or A|prime~, where A|prime~ is the value of A at the alternative location.(9) Given this Cobb-Douglas technology, the ratio of shadow prices in equation (3) is

|Pi~|prime~/|Pi~ = (A/A|prime~)|(h|prime~/h).sup.|Alpha~~|(p|prime~/p).sup.|Beta~~ |(w|prime~/w).sup.1-|Alpha~-|Beta~~. (6)

The derivation of equation (6) is shown in Appendix A.

The partial derivatives of R with respect to each relevant variable can now be found. The relevant variables include market variables at both locations such as the husband's income, Y and Y|prime~, the wife's wage rate, w and w|prime~, and housing prices, h and h|prime~; family characteristics such as the husband's age, the wife's age, the education of the wife, and the education of the husband; and regional amenities such as whether there are close friends or close relatives living at each location who can help the family in times of need (denoted as CFR and CFR|prime~).

The impact of the market and regional amenity variables on locational rent is straight forward. In looking at equations (3) and (6), it is obvious that |Delta~R/|Delta~Y |is greater than~ 0, |Delta~R/|Delta~Y|prime~ |is less than~ 0, |Delta~R/|Delta~h |is less than~ 0, |Delta~R/|Delta~p |is less than~ 0, |Delta~R/|Delta~p|prime~ |is greater than~ 0 and |Delta~R/|Delta~h|prime~ |is greater than~ 0. The signs of the partial derivatives for the wife's wage rate are not as immediately obvious. These derivatives are

|Delta~R/|Delta~w = (T + I(1 - |Alpha~ - |Beta~))|Pi~|prime~/|Pi~ |is greater than~ 0 (7)

and

|Delta~R/|Delta~w|prime~ = -I|prime~(1 - |Alpha~ - |Beta~)|Pi~|prime~/|Pi~ - T |is less than~ 0. (8)

The amenity variables are thought to increase household productivity at their respective locations. That is, close friends and relatives at the current location increase A, while close friends and relatives at the alternative location increase A|prime~. Hence, CFR will be positively related to R because |Delta~R/|Delta~A |is greater than~ 0 and CFR|prime~ will be negatively related to R because |Delta~R/|Delta~A|prime~ |is less than~ 0.

The education variables require more investigation because they could influence the income variables and household productivity at both current and alternative locations. Education is usually thought to be positively related to migration.(10) The husband's education affects locational rent through its impact on Y and Y|prime~ and the wife's education affects locational rent through its impact on w, w|prime~, A and A|prime~. A negative relationship between the husband's education and locational rent could occur if education raises the ratio Y|prime~/Y. Similarly, a negative relationship between the wife's education and locational rent could occur if more education raises the ratio w|prime~/w or A|prime~/A.

An explanation of why education might increase Y|prime~/Y, w|prime~/w, or A|prime~/A comes from the distinction between specific and general human capital. General human capital can be transferred to another job or region. Education is thought of as increasing general human capital whereas experience will increase both general and specific human capital. Y, w and A will be partly determined by specific human capital which cannot be transferred to another location. A higher ratio of general to specific human capital implies that a move will be less likely to lead to a loss in income, wages and household productivity. Since education increases the ratio of general to specific human capital, an increase in education might be expected to increase Y|prime~/Y, w|prime~/w and A|prime~/A and thereby, reduce locational rent. (See Appendix B for a mathematical treatment of the effects of education on locational rent.)

The impact of age on locational rent is similar to that of education in the sense that, at least for the ages considered in this study, age will increase human capital.(11) However, experience is a function of age and may increase specific human capital to a greater extent than it increases general human capital. Consequently, an increase in age would have an indeterminate effect on Y|prime~/Y, w|prime~/w and A|prime~/A and could either increase or decrease locational rent.(12)

IV. The Data and Empirical Model

The data used in this study are longitudinal data from the Panel Study of Income Dynamics. 1985 was the most recent year available when this study was undertaken. The families studied are a subset of the entire Panel Study set. Families in the subset consist of a husband and a wife who were married by 1980 and where the husband's age was less than fifty in 1980. Consequently, we are not likely to be looking at families which move because of retirement. For each year, migration can be identified by seeing whether the family changed their state or county of residence from the previous year. Migration only refers to joint moves by the husband and wife. Divorce or separation, if it occurs before the family moves, is treated as censoring the data.

Three types of migration are considered. These types of migration variables are called All Moves, Moves Within a State, and Moves Between States. These distinctions are made to provide some evidence as to the role of distance in family migration. Distance is usually thought to deter migration because relocation costs are positively associated with distance.(13) Distance can, however, affect more than just the initial costs of a move. The household's production technology becomes adapted to the current location. Families are familiar with relative prices and available stores, goods and amenities at their current and nearby locations. Thus, A|prime~/A will fall with distance. Families are also familiar with local business laws and regulations, and market opportunities, particularly labor market opportunities, in areas near their current location. This familiarity does decline with distance and consequently Y|prime~/Y and w|prime~/w will fall with distance. This expected impact of distance on A|prime~/A, Y|prime~/Y and w|prime~/w is by no means uniform. Centers of high market opportunities may be separated by intervening locations with fewer opportunities. However, on average, distance will play a role. Since distance plays a potential role in determining migration, separate estimates of moves within a state and moves between states are made to ascertain whether moves of different distances respond differently to the variables in the model.

The probability that a family will move, for each of the three types of moves, is estimated using a proportional hazard model.(14) A hazard model was chosen because it provides a convenient way of handling censored observations and because it can easily handle variables which may change with time. A hazard function gives the conditional probability that a family will move from its current location given that the family is still residing at that location.

The explanatory variables come from the theoretical model discussed in the previous section. Y|prime~, w|prime~ p, h|prime~ and p|prime~ are not available except possibly ex post for families which have moved.(15) In addition, h is not available but a variable called housing expenditures, HE (defined as HE = hH), is available. The explanatory variables used include the husband's income, Y; the wife's wage rate, w; the husband's education, |S.sub.h~; the wife's education, S; the family's expenditures on housing, HE; the close friends and relatives variables, CFR and CFR|prime~; and the ages of the husband and wife. Specifically, the husband's income is total family income minus the wife's earnings. The wife's wage rate is her hourly wage rate and is set at zero if she is not working. Education is in terms of levels of schooling. Age is in years. Housing expenditures are the family's monthly rent or housing payments for 1980. All of these variables, except HE, CFR, and CFR|prime~, are available for each year and, thus, are treated as time varying covariates.

The regional amenity variables, CFR and CFR|prime~, are dummy variables. CFR comes from the response to the following question: "Suppose there were a serious emergency in your household. Is there a friend or relative living nearby whom you could call on to spend a lot of time helping out?" If the answer is "yes," CFR = 1; otherwise, CFR = 0. CFR|prime~ is similarly defined from another question about help from someone not living near the family. These variables may represent much more than the obvious factor of emergency help. The nearness of close friends and relatives may influence many aspects of household production from entertainment, to child care, to chance conversations which provide an exchange of information and a feeling of closeness. Consequently, CFR would increase A in the household production function and

thus would be expected to have a negative relationship with family migration. CFR|prime~ would increase A|prime~ and consequently would be expected to be positively related to family migration. However, since it is not clear how to empirically identify the alternative location in a two region model when there are numerous regions, the interpretation of CFR|prime~ is ambiguous. The family might have CFR|prime~ = 1 and move, but move to a location where there are no close friends or relatives. Hence, the expected relationship between CFR|prime~ and migration might be weak at best. CFR|prime~ is included in the empirical estimates for symmetry with CFR. Its exclusion does not affect the results in any appreciable manner. Both CFR and CFR|prime~ are available only for 1980 which is why they are treated as fixed covariates.

Since the housing expenditures variable, HE, is not the variable, h, in the theoretical model, further discussion of HE is needed. Since HE = hH, |Delta~R/|Delta~HE will depend upon the demand elasticity for housing. Recall that |Delta~R/|Delta~h |is less than~ 0, and hence h is positively related to migration. If the demand for housing is inelastic, then as h changes, HE will move in the same direction. Thus, both h and HE would be positively related to migration. If the demand for housing is elastic, HE will be negatively related to migration. Since housing demand is usually found to be inelastic, HE should be positively related to migration.(16)

V. The Empirical Results

The Probability of Migration

Hazard model estimates of the probability a family will migrate are shown in Table I. The estimates for two different equations are shown for each type of move. In columns (a), (c) and (e), all the variables discussed in the previous section are included in the model. In columns (b), (d) and (f), only those variables which were significant in column (a) are used. In general, the results are better for all moves, columns (a) and (b), and for moves between states, columns (e) and (f), than for moves within a state, columns (c) and (d). The results, however, are similar for the three types of moves. No statistically significant variable changes sign. Since the determinants of family migration are similar for the three types of moves, specific reference to the type of move will add little to the general discussion of the results in Table I. We will refer to column (a) when discussing the empirical results.

The signs of the coefficients are as anticipated. The weakest results are for CFR|prime~, the husband's education, |S.sub.h~, and the age variables. These variables are insignificant. The insignificance of the husband's education, the husband's age, and the wife's age may reflect problems with colinearity. In estimates not shown here, the exclusion of the wife's education, S, makes the husband's education significant. In addition, when one of the age variables was excluded, the other age variable becomes significant. CFR|prime~, however, is insignificant in all the models which were estimated. CFR|prime~ was expected to have a positive sign because the presence of close friends and relatives at alternative locations is expected to increase A|prime~ decreasing locational rent and hence increasing the probability of a move. Recall, however, that there was no way of knowing whether TABULAR DATA OMITTED a family actually moved to a location where there are close friends and relatives. Consequently, the insignificance of this variable is not surprising.

While the education variables, S and |S.sub.h~, have a positive relationship with family migration, only the wife's education is significant. Recall that the explanation for this positive sign rests on the notion that education represents general human capital. Hence, more education would increase Y|prime~/Y, for the husband's education, and would increase both w|prime~/w and A|prime~/A, for the wife's education. The significance and larger value of the estimated coefficient for the wife's education (than for the husband's education) supports the notion that there is an additional channel, A|prime~/A, through which the wife's education influences migration.

The remaining variables are both significant and have the expected signs. Since the theoretical model has clear implications for the signs of four of these variables, Y, w, CFR and HE, the results confirm the model. The husband's income, Y, has the expected negative impact on migration as does w, the wife's wage rate. Higher husband's income at the current location, ceteris paribus, makes the current location more attractive and reduces the probability the family will move. A higher wife's wage rate also increases the attractiveness of the current location and reduces family migration. Recall, that this negative impact of the wife's wage rate on migration is central to the tied-mover hypothesis. CFR has the expected negative sign. Recall that CFR was TABULAR DATA OMITTED expected to have a negative sign because the presence of close friends and relatives at the current location enhances household production, increases A, and, consequently, increases the family's locational rent reducing the probability the family will move to another region. Finally, the housing expenditures variable, HE, has a positive and significant relationship with family migration. As previously discussed, a positive relationship is expected because independent empirical studies have found an inelastic demand for housing.

The Distance of Migration

Having discussed which families are most likely to move, we will now consider whether the move was within a state or between states. Recall that Y|prime~, and w|prime~ fall with distance and that |Pi~|prime~ rises with distance because the family's market earning potential and its household technology will be adapted to its current surroundings. Letting D represent distance, |Delta~I|prime~/|Delta~D and |Delta~|Pi~|prime~/|Delta~D will depend upon the same variables which influence locational rent. Variables, like education, which are related to general human capital will make the family more adaptable to a variety of locations and thereby increase the probability that a move will be to another state. Variables, such as income and wage rates, which are location specific are likely to be negatively related to the probability that a move will be to another state.

Table II presented the logit estimates of the conditional probability that a family will move across state boundaries, given that the family moved. Of the 469 families that moved, 272 moved to another state and 197 moved to another county in their state. Estimates of the conditional probability of a move to another state are based on the 1980 value of variables defined at the original location. Only variables with a t-statistic greater than one are included in the final model. These variables are the husband's income, the wife's wage rate, housing expenditures, the wife's education and the husband's age.

In Table II, the husband's income and the wife's wage rate show a negative impact on the probability that a move was to another state. This negative impact is consistent with the notion that these variables partly reflect location specific human capital. The value of this human capital diminishes with distance. The education of the wife and the husband's age have a positive impact on the probability that a move was to another state. This positive impact is consistent with the notion that both the wife's education and the husband's age reflect general human capital. Finally, housing expenditures have a positive impact on the probability that a move will be between states. This positive impact would seem to indicate that higher priced housing is more adaptable to distant locations than is lower priced housing. This difference in adaptability might be due to less information about the quality of lower priced housing. Familiarity with the housing market diminishes with distance and this familiarity may be more important for families with lower housing expenditures.

VI. Conclusions

The theoretical model of family migration which was developed in this paper stresses the microeconomics of location choice. Notions of regional amenities and psychic costs can be given concrete interpretations within the framework of a Becker-Lancaster model of household production. Empirical hypotheses are derived from the comparative static model. An advantage of this approach is that these empirical hypotheses have straightforward interpretations within the familiar framework of choice theory. Furthermore, the empirical results support the hypotheses derived from the model. In this concluding section, we will first draw some general conclusions about the empirical results. Then we will discuss the policy implications both of the model and of the empirical results.

Both the estimates of the probability of migration and of the distance of the move are consistent with the predictions of the theoretical model. As predicted by the model, the earnings variables, which are linked to the current location, have a negative impact both on family migration and on the distance of the move. Also, as predicted by the model, variables which influence migration through their impact on general human capital, specifically the education variables, have a positive impact on family migration and on the distance of the move. That the wife's education variable was dominant over the husband's education variable suggests that there is some importance of the wife's education in household production. The age variables, although mostly reflecting indeterminacy, are positive and significant in predicting longer distanced moves. Variables related to location specific amenities, the location of close friends and relatives variables, also had the predicted signs. Finally, housing expenditures were predicted to have a positive impact on migration if the demand for housing is inelastic. Housing expenditures also had a positive impact on the distance of a move suggesting that information about the quality of less expensive housing falls with distance to a greater extent than for more expensive housing. More expensive housing is therefore easier to relocate to unfamiliar regions.

The model does have a fairly general policy implication. Families are viewed as living at their optimal location. Hence, migration is not viewed as the slow response to disequilibrium in the labor market. Consequently, increasing the rate of migration would not, per se, lead to gains in social welfare for the economy as a whole.(17) Some productivity gains could occur, through migration to regions with a higher marginal productivity of labor in producing goods. However, there would be a greater loss in the productivity of time in household production. Productivity gains could still be achieved through moving capital to regions with a lower marginal productivity of labor in producing goods. Hence, the model is consistent with the view that regional development programs are desirable.

Another policy implication of the model concerns labor market programs. Typically, job placement programs, both public and private, concentrate on locating employment for individuals. Families, however, are concerned with the employment of both husband and wife and with the household production of the family. Hence, families may be reluctant to move to unfamiliar labor markets on the basis of a job offer for the husband or wife alone. They will only move, if they believe it will be fairly easy to find suitable employment for the tied mover in the alternative location and to adjust household production to the alternative location. The positive signs of the wife's education and the husband's age in Table II support this notion. General human capital makes it more likely that a family will move to a less familiar region. A greater emphasis of employment programs on the joint employment and location decisions of the husband and wife could improve the efficiency of the labor market.

Appendix A: Finding the Shadow Price

The production functions for the two locations are

Z = A|H.sup.|Alpha~~|X.sup.|Beta~~|t.sup.1-|Alpha~-|Beta~~ (A.1)

and

Z = A|prime~|H.sup.|Alpha~~|X.sup.|Beta~~|t.sup.1-|Alpha~-|Beta~~. (A.2)

|Theta~ and L will influence household production through their impact on A and A|prime~. When t |is less than~ T, the wife is employed and the budget constraint for commodities at each location can be written as

I = Y + wT = |Pi~Z, (A.3)

for the current location, and as

I|prime~ = Y|prime~ + w|prime~T = |Pi~|prime~Z, (A.4)

for the alternative location.

To find the shadow price, |Pi~, in (A.3) note that the budget for goods can be written as

I = Y + wT = hH + pX + wt. (A.5)

The first step in finding |Pi~ is to solve for H and X as functions of t. Optimal production of commodities implies that the household will use inputs up to the point where

(|Delta~Z/|Delta~X)/p = (|Delta~Z/|Delta~t)/w

and

(|Delta~Z/|Delta~H)/h = (|Delta~Z/|Delta~t)/w. (A.6)

By differentiating (A.1), simplifying, and substituting into (A.6), X and H can be found as functions of t yielding,

X = (w/p)(|Beta~/(1 - |Alpha~ - |Beta~))t

and

H = (w/h)(|Alpha~/(1 - |Alpha~ - |Beta~))t. (A.7)

Note that (A.7) implies that X/t and H/t have unique solutions determined by the production elasticities and by w, p, and h.

The second step in finding |Pi~ is to express t as a function of Z. Taking advantage of linear homogeneity and dividing (A.1) by t yields

Z/t = A||H/t~.sup.|Alpha~~||X/t~.sup.|Beta~~. (A.8)

Then substituting X/t and H/t, from (A.7), into (A.8), yields

Z/t = A||(w/h)(|Alpha~/(1 - |Alpha~ - |Beta~))~.sup.|Alpha~~||(w/p)(|Beta~/(1 - |Alpha~ - |Beta~))~.sup.|Beta~~. (A.9)

We can now find |Pi~ by solving (A.9) for t, substituting for t in (A.7) to find H, X, and t as functions of Z. Substituting these expressions for H, X and t into the budget constraint (A.5) and simplifying yields |Pi~Z, where

|Pi~ = w/A(1 - |Alpha~ - |Beta~)||(w/h)(|Alpha~/(1 - |Alpha~ - |Beta~))~.sup.|Alpha~~||(w/p)(|Beta~/(1 - |Alpha~ - |Beta~))~.sup.|Beta~~. (A.10)

Similarly, for the alternative location,

|Pi~|prime~ = w|prime~/A|prime~(1 - |Alpha~ - |Beta~)||(w|prime~/h|prime~)(|Alpha~/(1 - |Alpha~ - |Beta~))~.sup.|Alpha~~||(w|prime~/p|prime~)(|Beta~/(1 - |Alpha~ - |Beta~))~.sup.|Beta~~. (A.11)

Hence, the ratio of shadow prices |Pi~|prime~/|Pi~ can be found by dividing (A.10) into (A.11) and simplifying, yielding

|Pi~|prime~/|Pi~ = (A/A|prime~)|(h|prime~/h).sup.|Alpha~~|(p|prime~/p).sup.|Beta~~ |(w|prime~/w).sup.1-|Alpha~-|Beta~~. (A.12)

When the wife specializes in household production, t = T, the commodity budget constraints are nonlinear and become

I = Y = q|Z.sup.1/(|Alpha~+|Beta~)~ and

I|prime~ = Y|prime~ = q|prime~|Z.sup.1/(|Alpha~+|Beta~)~. (A.13)

The same approach as before yields

q = {p(|Alpha~ + |Beta~)/|Beta~}/{|A.sup.1/(|Alpha~+|Beta~)~|T.sup.(1-|Alpha~-|B eta~)/(|Alpha~+|Beta~)~||(p/h)(|Alpha~/|Beta~)~.sup.|Alpha~/(|A lpha~+|Beta~)~}, (A.14)

and

q|prime~ = {p|prime~(|Alpha~ + |Beta~)/|Beta~}/{|A|prime~.sup.1/|Alpha~+|Beta~)~|T.sup.(1-|Alp ha~-|Beta~)/(|Alpha~+|Beta~)~||(p|prime~/h|prime~)(|Alpha~/|Bet a~)~.sup.|Alpha~/(|Alpha~+|Beta~)~}. (A.15)

Dividing (A.14) into (A.15) yields, upon simplification,

q|prime~/q = ||(A/A|prime~)|(h|prime~/h).sup.|Alpha~~|(p|prime~/p).sup.|Beta ~~~.sup.1/(|Alpha~+|Beta~)~. (A.16)

Appendix B: Education and Locational Rent

To see the impact of the husband's education, |S.sub.h~, on locational rent, first note that

|Delta~(Y|prime~/Y)/|Delta~|S.sub.h~ = (|Delta~Y|prime~/|Delta~|S.sub.h~)/Y - (Y|prime~/|Y.sup.2~)(|Delta~Y/|Delta~|S.sub.h~). (B.1)

Next, differentiate R with respect to |S.sub.h~ yielding

|Delta~R/|Delta~|S.sub.h~ = (|Delta~Y/|Delta~|S.sub.h~)(|Pi~|prime~/|Pi~) - |Delta~Y|prime~/|Delta~|S.sub.h~. (B.2)

Rearranging terms in (B.1) and solving for |Delta~Y|prime~/|Delta~|S.sub.h~ and then substituting into (B.2) yields

|Delta~R/|Delta~|S.sub.h~ = |(|Delta~Y/|Delta~|S.sub.h~)(|Pi~|prime~/|Pi~ - Y|prime~/Y)~ - |Y(|Delta~(Y|prime~/Y)/|Delta~|S.sub.h~)~. (B.3)

To interpret equation (B.3), note that since |Delta~Y/|Delta~|S.sub.h~ |is greater than~ 0, the term in the first brackets will be positive if

(|Pi~|prime~/|Pi~ - Y|prime~/Y) |is greater than~ 0. (B.4)

Recall that, since the current location is, by assumption, the optimum location, R |is greater than~ 0. Hence, rearranging terms in equation (3) yields

(|Pi~|prime~/|Pi~ - I|prime~/I) |is greater than~ 0. (B.5)

Since husband's income is a component of full income, we would expect (B.4) to usually hold when (B.5) holds making the sign of the first bracketed term often positive. The usual argument that education represents general human capital will make |Delta~(Y|prime~/Y)/|Delta~|S.sub.h~ positive and hence make the second bracketed term positive. A positive second bracketed term could result in the husband's education having a negative impact on locational rent and, hence, a positive impact on migration.

A similar argument can be applied to the wife's education, S. The impact of the wife's education on locational rent is

|Delta~R/|Delta~S = |(|Delta~w/|Delta~S)T(|Pi~|prime~/|Pi~ - w|prime~/w~

- |(wT(|Delta~Y|prime~/Y)/|Delta~S)(1 - (1 - |Alpha~ - |Beta~)|Pi~|prime~/|Pi~(w|prime~/w))~

- ||Delta~(A/A|prime~)/|Delta~S)(IA|prime~|Pi~|prime~/A|Pi~)~. (B.6)

The term in the third brackets reflects the impact of the wife's education on household production. Uncertainty as to the sign of this term complicates the discussion. However, the same argument as with the husband's education applies to the first bracketed expression. The second bracketed expression is also likely to be positive. It will be positive unless

|Pi~|prime~/|Pi~ - (w|prime~/w)(1/(1 - |Alpha~ - |Beta~)) |is greater than~ 0. (B.7)

Since 1/(1 - |Alpha~ - |Beta~) |is greater than~ 1, inequality (B.7) would imply considerable regional differences in the costs of household production. In this situation, the wife's education could be positively associated with locational rent. Otherwise, |Delta~R/|Delta~S is likely to be negative.

Much of the theoretical part of this paper was written while on sabbatical at the University of Essex. We wish to thank the anonymous referee for useful suggestions about both the content and the organization of the paper.

1. Others looking at the wife's income and migration include Long |22~, Mincer and Polachek |27~, DaVanzo |7~, Polachek and Horvath |28~, Sandell |33~, and Bartel |1~.

2. While household production models have been applied to many areas of household choice including fertility, health care, marriage and divorce, these models have only recently received attention in the migration literature. See Shields and Shields |39~ for an analytical survey of the migration literature which includes a section on household production and migration.

3. Housing costs play an important role in hedonic wage models which stress that high wage rates or low housing costs are compensation for poor regional amenities. See Rosen |32~, Roback |30; 31~, Graves |10~, Blomquist, Berger and Hoehn |3~ and Knapp and Graves |17~.

4. In this model, the wife's time plays a dual role. It can be used in household production or in earning income. Note that the husband's time does not appear in the household production function. Thus, income can be divided into a component which is independent of the time used in household production, called the husband's income (Y), and a component, called the wife's earnings, w(T - t), which is partly determined by the time used in household production, t. While this treatment of time is not crucial for the migration model which will be developed, it is made because we wish to emphasize the labor market decisions of the tied mover.

5. If the wife completely specializes in household production, the implicit constraint for commodities is nonlinear. See Willis |41~ and Pollak and Wachter |29~.

6. For a discussion, see Greenwood |12~.

7. For a similar argument, expressed in terms of labor market search and the reservation wage rate, see Denslow and Eaton |8~. They express their argument in terms of the reservation wage rate at the alternative location for moving to that location.

8. Since there is only one aggregate commodity, U(Z) is maximized when Z is maximized if |Delta~U/|Delta~Z |is greater than~ 0 for all Z.

9. For a consideration of locations being chosen because they favor the production of certain commodities, see T. P. Schultz |34~.

10. A positive relationship between education and individual migration has usually been found. See Greenwood |11~, Bowles |4~, Long |21~, and Fields |9~. Explanations have included the idea that education will reduce the risks of a move either because education creates more employment opportunities |37~ or because education contributes to an individual's ability to collect and process information |35; 36~.

11. The maximum age of the husband for any year of this study is fifty-five. See Mincer |25~ for a discussion of the affects of both age and education on human capital.

12. See Bowles |4~ and Schwartz |37~ for a discussion of age and individual migration.

13. See Levy and Wadycki |20~, Schwartz |36~, Denslow and Eaton |8~, and Clark and Cosgrove |5~.

14. The hazard model was estimated using BMDP |15~. For a discussion of the hazard model see Cox |6~, Heckman and Singer |13~, and Tony Lancaster |19~. For the logit model, see Madalla |23~.

15. The exclusion of these variables will lead to biased estimates if they are correlated with variables in the estimated model. For example, Y and Y|prime~ would be correlated. However, since they have opposite effects on migration, the bias would be towards the null hypothesis.

16. See Houthakker and Taylor |16~ and Mansur and Whalley |24~ for studies which found an inelastic demand for housing.

17. See Herzog and Schlottman |14~ for a treatment of migration, allocative efficiency and the role of psychic versus informational and physical moving costs.

References

1. Bartel, A. P., "The Migration Decision: What Role Does Job Mobility Play?" American Economic Review, December 1979, 775-86.

2. Becker, Gary S., "A Theory of the Allocation of Time." Economic Journal, September 1965, 493-517.

3. Blomquist, Glenn; Mark Berger, and John Hoehn, "New Estimates of the Quality of Life in Urban Areas." American Economic Review, March 1988, 89-107.

4. Bowles, Samuel, "Migration as Investment: Empirical Tests of the Human Investment Approach to Geographical Mobility." Review of Economics and Statistics, November 1970, 356-62.

5. Clark, David E. and James C. Cosgrove, "Amenities Versus Labor Market Opportunities: Choosing the Optimal Distance to Move." Journal of Regional Science, August 1991, 311-28.

6. Cox, D. R., "Regression Models and Life-Tables." Journal of the Royal Statistical Society (Series B), 1972, 187-202.

7. DaVanzo, Julie. "Why Families Move: A Model of the Geographic Mobility of Married Couples." R-1972-DOL. Santa Monica, California: Rand Corporation, 1976.

8. Denslow, David A., Jr., and Peter J. Eaton, "Migration and Intervening Opportunities." Southern Economic Journal, October 1984, 369-87.

9. Fields, Gary S., "Place to Place Migration in Colombia." Economic Development and Cultural Change, April 1982, 539-58.

10. Graves, Philip E., "Migration with a Composite Amenity: The Role of Rents." Journal of Regional Science, November 1983, 541-46.

11. Greenwood, Michael J., "An Analysis of the Determinants of Geographic Labor Mobility in the United States." Review of Economics and Statistics, May 1969, 189-94.

12. -----, "Research on Internal Migration in the United States: A Survey." Journal of Economic Literature, June 1975, 397-433.

13. Heckman, James J. and Burton Singer. "Social Science Duration Analysis," in Longitudinal Analysis of Labor Market Data, edited by J. J. Heckman and B. Singer. Cambridge: Cambridge University Press, 1985, pp. 39-110.

14. Herzog, Henry W., Jr. and Alan M. Schlottmann, "Labor Force Migration and Allocative Efficiency in the United States: The Roles of Information and Psychic Costs." Economic Inquiry, July 1981, 459-75.

15. Hopkins, Alan. "Survival Analysis with Covariates--Cox Models," in BMDP Statistical Software Manual, Volume 2, edited by W. J. Dixon. Berkeley: University of California Press, 1988, pp. 719-43.

16. Houthakker, Hendrik S. and Lester D. Taylor, Consumer Demand in the United States, 1929-1970: Analysis and Projections. Cambridge: Harvard University Press, 1966.

17. Knapp, Thomas A. and Philip E. Graves, "On the Role of Amenities in Models of Migration and Regional Development." Journal of Regional Science, February 1989, 71-87.

18. Lancaster, Kelvin J., "A New Approach to Consumer Theory." Journal of Political Economy, April 1966, 132-57.

19. Lancaster, Tony. The Econometric Analysis of Transition Data. Cambridge: Cambridge University Press, 1990.

20. Levy, Mildred B. and Walter J. Wacycki, "The Influence of Family and Friends on Geographic Labor Mobility: An International Comparison." Review of Economics and Statistics, May 1973, 198-203.

21. Long, Larry H., "Migration Differentials by Education and Occupation: Trends and Variations." Demography, May 1973, 243-58.

22. -----, "Women's Labor Force Participation and the Residential Mobility of Families." Social Forces, March 1974, 342-48.

23. Madalla, G. S. Limited-Dependent and Qualitative Variables in Econometrics. Cambridge: Cambridge University Press, 1983.

24. Mansur, Ahsan and John Whalley. "Numerical Specification of Applied General Equilibrium Models: Estimation, Calibration, and Data," in Applied General Equilibrium Analysis, edited by H. E. Scarf and J. B. Shoven. New York: Cambridge University Press, 1984, p. 109.

25. Mincer, Jacob. Schooling, Experience, and Earnings. New York: National Bureau of Economic Research, Columbia University Press, 1974.

26. -----, "Family Migration Decisions." Journal of Political Economy, October 1978, 749-73.

27. ----- and Solomon Polachek, "Family Investments in Human Capital: Earnings of Women." Journal of Political Economy, April Supplement 1974, 76-108.

28. Polachek, Solomon W. and Francis W. Horvath. "A Life Cycle Approach to Migration: Analysis of the Perspicacious Peregrinator," in Research in Labor Economics, edited by R. G. Ehrenberg. Greenwich, Connecticut: J.A.I. Press, 1977, pp. 103-42.

29. Pollak, Robert A. and Michael L. Wachter, "The Relevance of the Household Production Function for the Allocation of Time." Journal of Political Economy, April 1975, 255-77.

30. Roback, Jennifer, "Wages, Rents, and the Quality of Life." Journal of Political Economy, December 1982, 1257-78.

31. -----, "Wages, Rents, and Amenities: Differences Among Workers and Regions." Economic Inquiry, January 1988, 23-41.

32. Rosen, Sherwin. "Wage-Based Indexes of Urban Quality of Life," in Current Issues in Urban Economics, edited by P. Mieszkowski and M. Straszheim. Baltimore: Johns Hopkins Press, 1979, pp. 74-104.

33. Sandell, Steven H., "Women and the Economics of Family Migration." Review of Economics and Statistics, November 1977, 406-14.

34. Schultz, T. Paul. "Heterogeneous Preferences and Migration Self-Selection, Regional Prices and Programs, and the Behavior of Migrants in Colombia," in Research in Labor Economics, edited by T. P. Schultz. Greenwich Connecticut: J.A.I. Press, 1988, pp. 163-81.

35. Schultz, Theodore W., "The Value of the Ability to Deal with Disequilibria." Journal of Economic Literature, September 1975, 827-46.

36. Schwartz, Aba, "Interpreting the Effect of Distance on Migration." Journal of Political Economy, September/October 1973, 1153-69.

37. -----, "Migration, Age, and Education." Journal of Political Economy, August 1976, 701-19.

38. Shields, Gail M. and Shields, Michael P., "Family Migration and Nonmarket Activities in Costa Rica." Economic Development and Cultural Change, October 1989, 73-88.

39. ----- and -----, "The Emergence of Migration Theory and a Suggested New Direction." Journal of Economic Surveys, 1989, 277-304.

40. Sjaastad, Larry A., "The Costs and Returns of Human Migration." Journal of Political Economy, October supplement 1962, 80-93.

41. Willis, Robert J., "A New Approach to the Economic Theory of Fertility Behavior." Journal of Political Economy, March/April supplement 1973, 514-64.

Migration studies have usually focussed on individual persons or workers. There has, however, been some interest in family migration. While single individuals in the labor market only have to worry about their own income, families, where both the husband and wife may be in the labor market, need to be concerned with the income of both spouses. Hence, double income families may be less likely to move than single income families. Jacob Mincer |26~ develops and explores this hypothesis, to be called the tied-mover hypothesis.(1) The tied-mover hypothesis has recently been extended to include not only considerations of the wife's earnings but also considerations of her household production as factors in the family's migration decisions |38~. In this paper, the research on family migration will be extended in three ways. A formal model of household production and family migration will be developed and used to generate testable hypotheses about family migration. Variables, which otherwise would not have a clear economic interpretation, are incorporated into the model. The model will then be estimated and some policy implications will be discussed. One policy implication concerns the efficiency of the labor market. Labor market efficiency needs to be judged not only in terms of the efficiency of the job search of single individuals but also in terms of the efficiency of the joint job search of double income families.

This paper is organized into five sections. First, a comparative static household production model of locational choice is developed.(2) Next, this model of locational choice is used to develop a theoretical model of family migration. Third, the data and empirical model are discussed. Fourth, the empirical results are reported and interpreted in terms of three different types of migration. Estimates of the impact of selected variables on the distance of the move are also considered. The final section summarizes the general findings and offers some policy implications.

II. A Model of Locational Choice

In this section, a comparative static model of family migration is developed using a Becker-Lancaster household production approach |2; 18~. The family will be assumed to live at its optimum location. The technology of the household and market variables, such as prices, the husband's income and the wife's wage rate, will be assumed to depend upon the family's location. The family moves when these location-specific variables change in a way which makes the current location no longer optimal. We will see, in section V, that this approach is capable of explaining the patterns of family migration which are observed.

In a standard household production model, the household receives utility directly from the commodities which it consumes. Commodities are produced from inputs which consist of goods purchased in the market and the household's own time. A household maximizes its utility from consuming commodities subject to a linear income constraint and the household's technology. This model can be applied to migration by introducing, into the household's production functions, characteristics of family members and of the location of the household. Consider an example where a household, with both a husband and a wife, maximizes

U = U(|Z.sub.1~,|Z.sub.2~,...|Z.sub.n~) (1a)

subject to

B = Y + w(T - |summation of~ (|t.sub.i~) where i = 1 to n) = |summation of~ (|p.sub.j~|X.sub.j~) where j = 1 to m + hH, (1b)

and

|Z.sub.i~ = |f.sub.i~(|X.sub.1i~,|X.sub.2i~,...|X.sub.mi~;H;|t.sub.i~/|Thet a~,L), for i = 1,2,...n, (1c)

where

|X.sub.j~ = |summation of~ (|X.sub.ji~) where i = 1 to n, for j = 1,2,...m. (1d)

U is family utility, |Z.sub.i~ is the ith commodity, |f.sub.i~ is the household production function for the ith commodity, |X.sub.ji~ is the quantity of good j used to produce commodity i, |p.sub.j~ is the price of good j, |t.sub.i~ is the wife's time used to produce commodity i, w is the market wage rate of the wife, T is the total time the wife has at her disposal, Y is the husband's income which includes non-wage family income, B is the family budget, h is the price of housing services, H is housing services, |Theta~ is a vector of household characteristics, and L is a vector of relevant attributes of the family's current location.(3) Note that some household characteristics such as education will influence both the household production function, and the earnings variables, w and Y.(4)

Many of the exogenous variables in the above constraints depend upon the family's location and will change if the family moves to another location. Location will influence family utility in two ways. First, market prices, housing costs, the husband's income and the wife's wage rate depend upon location. Second, location will have a direct effect on household production possibilities because elements in L, which can differ from one location to the next, are arguments in the production functions of the household. Included in L are such factors as climate, proximity to friends and relatives, and the availability of cultural, recreational and educational amenities. Note that the impact that location has on Y, w, and production possibilities will be affected by household characteristics, |Theta~, such as marital status, the number of children, the ages of family members, and the education of family members.

If the production functions are linearly homogeneous and the wife does not completely specialize in household production, then the optimal production of commodities implies that the constraints in (1) can be combined into a single, linear constraint,

I = |Pi~Z, (2)

where I = Y + wT is called full income, |Pi~ is a vector of implicit (shadow) prices of commodities and Z is a vector of commodities.(5) The shadow prices depend upon Y, w, h, |Theta~, L, and a vector of market prices, p.

In comparing locations, families, which are living at their optimal location, receive rent in the sense that there is some amount of money which would be just sufficient to induce them to move |40~.(6) This implicit rent will be called locational rent. To more concretely define locational rent, suppose there are only two locations. Let I and I|prime~ be full income at each location, and let |Pi~ and |Pi~|prime~ be scalars representing the cost of commodity production at each location. Since locational rent would equate real commodity consumption at both locations, locational rent can be written as

R = I(|Pi~|prime~/|Pi~) - I|prime~, (3)

where R is locational rent.

To illustrate the role locational rent plays in locational choice, consider an example where there are two periods of time. At the beginning of the first period, the family is assumed to be living at its optimal location. Hence, initial locational rent is positive, given the vector of exogenous variables governing the family's choice for that period. At the end of the period, the family will face a new vector of exogenous variables and, hence, would receive a new level of locational rent. If this new level of locational rent is negative, the family moves.

A basic assumption of the migration model developed here is that the larger the locational rent, the less likely any given change in the family's economic situation will be of sufficient magnitude to cause a move. For example, an increase in the income that the husband could earn at another location will reduce locational rent but will not necessarily cause the family to move. The increase in income must exceed the original level of locational rent if the family is to move.(7)

III. The Migration Model

Since household migration is assumed to be a function of locational rent, where locational rent is a function of market variables, household characteristics, and locational characteristics, the impact that these variables have on migration can be discussed in terms of how they affect locational rent. The basic underlying model is

M = f(R), (4)

where M is the probability that the family will move by the end of a given period and R is locational rent at the beginning of the period.

It is by no means straight forward to estimate equation (4). Three difficulties arise. First, there are more than two locations. Second, the market variables for the family at an alternative location, denoted as Y|prime~, p|prime~, and w|prime~, are not known to the investigator. Third, the model is intrinsically nonlinear. The model does, however, have some clear implications as to the direction of the impact which the known variables have on migration.

The direction of this impact can be found by differentiating R with respect to the relevant variables. To do so, consider a simple example where there is only one aggregate commodity produced by a Cobb-Douglas production function

Z = A|H.sup.|Alpha~~|X.sup.|Beta~~|t.sup.1-|Alpha~-|Beta~~ (5)

with inputs of market goods, the wife's time, and housing.(8) Furthermore, assume there are only two locations; the optimal current location and a suboptimal alternative location. Finally, to simplify the discussion of regional differences in household technology, assume that the only difference in these technologies is the value of A or A|prime~, where A|prime~ is the value of A at the alternative location.(9) Given this Cobb-Douglas technology, the ratio of shadow prices in equation (3) is

|Pi~|prime~/|Pi~ = (A/A|prime~)|(h|prime~/h).sup.|Alpha~~|(p|prime~/p).sup.|Beta~~ |(w|prime~/w).sup.1-|Alpha~-|Beta~~. (6)

The derivation of equation (6) is shown in Appendix A.

The partial derivatives of R with respect to each relevant variable can now be found. The relevant variables include market variables at both locations such as the husband's income, Y and Y|prime~, the wife's wage rate, w and w|prime~, and housing prices, h and h|prime~; family characteristics such as the husband's age, the wife's age, the education of the wife, and the education of the husband; and regional amenities such as whether there are close friends or close relatives living at each location who can help the family in times of need (denoted as CFR and CFR|prime~).

The impact of the market and regional amenity variables on locational rent is straight forward. In looking at equations (3) and (6), it is obvious that |Delta~R/|Delta~Y |is greater than~ 0, |Delta~R/|Delta~Y|prime~ |is less than~ 0, |Delta~R/|Delta~h |is less than~ 0, |Delta~R/|Delta~p |is less than~ 0, |Delta~R/|Delta~p|prime~ |is greater than~ 0 and |Delta~R/|Delta~h|prime~ |is greater than~ 0. The signs of the partial derivatives for the wife's wage rate are not as immediately obvious. These derivatives are

|Delta~R/|Delta~w = (T + I(1 - |Alpha~ - |Beta~))|Pi~|prime~/|Pi~ |is greater than~ 0 (7)

and

|Delta~R/|Delta~w|prime~ = -I|prime~(1 - |Alpha~ - |Beta~)|Pi~|prime~/|Pi~ - T |is less than~ 0. (8)

The amenity variables are thought to increase household productivity at their respective locations. That is, close friends and relatives at the current location increase A, while close friends and relatives at the alternative location increase A|prime~. Hence, CFR will be positively related to R because |Delta~R/|Delta~A |is greater than~ 0 and CFR|prime~ will be negatively related to R because |Delta~R/|Delta~A|prime~ |is less than~ 0.

The education variables require more investigation because they could influence the income variables and household productivity at both current and alternative locations. Education is usually thought to be positively related to migration.(10) The husband's education affects locational rent through its impact on Y and Y|prime~ and the wife's education affects locational rent through its impact on w, w|prime~, A and A|prime~. A negative relationship between the husband's education and locational rent could occur if education raises the ratio Y|prime~/Y. Similarly, a negative relationship between the wife's education and locational rent could occur if more education raises the ratio w|prime~/w or A|prime~/A.

An explanation of why education might increase Y|prime~/Y, w|prime~/w, or A|prime~/A comes from the distinction between specific and general human capital. General human capital can be transferred to another job or region. Education is thought of as increasing general human capital whereas experience will increase both general and specific human capital. Y, w and A will be partly determined by specific human capital which cannot be transferred to another location. A higher ratio of general to specific human capital implies that a move will be less likely to lead to a loss in income, wages and household productivity. Since education increases the ratio of general to specific human capital, an increase in education might be expected to increase Y|prime~/Y, w|prime~/w and A|prime~/A and thereby, reduce locational rent. (See Appendix B for a mathematical treatment of the effects of education on locational rent.)

The impact of age on locational rent is similar to that of education in the sense that, at least for the ages considered in this study, age will increase human capital.(11) However, experience is a function of age and may increase specific human capital to a greater extent than it increases general human capital. Consequently, an increase in age would have an indeterminate effect on Y|prime~/Y, w|prime~/w and A|prime~/A and could either increase or decrease locational rent.(12)

IV. The Data and Empirical Model

The data used in this study are longitudinal data from the Panel Study of Income Dynamics. 1985 was the most recent year available when this study was undertaken. The families studied are a subset of the entire Panel Study set. Families in the subset consist of a husband and a wife who were married by 1980 and where the husband's age was less than fifty in 1980. Consequently, we are not likely to be looking at families which move because of retirement. For each year, migration can be identified by seeing whether the family changed their state or county of residence from the previous year. Migration only refers to joint moves by the husband and wife. Divorce or separation, if it occurs before the family moves, is treated as censoring the data.

Three types of migration are considered. These types of migration variables are called All Moves, Moves Within a State, and Moves Between States. These distinctions are made to provide some evidence as to the role of distance in family migration. Distance is usually thought to deter migration because relocation costs are positively associated with distance.(13) Distance can, however, affect more than just the initial costs of a move. The household's production technology becomes adapted to the current location. Families are familiar with relative prices and available stores, goods and amenities at their current and nearby locations. Thus, A|prime~/A will fall with distance. Families are also familiar with local business laws and regulations, and market opportunities, particularly labor market opportunities, in areas near their current location. This familiarity does decline with distance and consequently Y|prime~/Y and w|prime~/w will fall with distance. This expected impact of distance on A|prime~/A, Y|prime~/Y and w|prime~/w is by no means uniform. Centers of high market opportunities may be separated by intervening locations with fewer opportunities. However, on average, distance will play a role. Since distance plays a potential role in determining migration, separate estimates of moves within a state and moves between states are made to ascertain whether moves of different distances respond differently to the variables in the model.

The probability that a family will move, for each of the three types of moves, is estimated using a proportional hazard model.(14) A hazard model was chosen because it provides a convenient way of handling censored observations and because it can easily handle variables which may change with time. A hazard function gives the conditional probability that a family will move from its current location given that the family is still residing at that location.

The explanatory variables come from the theoretical model discussed in the previous section. Y|prime~, w|prime~ p, h|prime~ and p|prime~ are not available except possibly ex post for families which have moved.(15) In addition, h is not available but a variable called housing expenditures, HE (defined as HE = hH), is available. The explanatory variables used include the husband's income, Y; the wife's wage rate, w; the husband's education, |S.sub.h~; the wife's education, S; the family's expenditures on housing, HE; the close friends and relatives variables, CFR and CFR|prime~; and the ages of the husband and wife. Specifically, the husband's income is total family income minus the wife's earnings. The wife's wage rate is her hourly wage rate and is set at zero if she is not working. Education is in terms of levels of schooling. Age is in years. Housing expenditures are the family's monthly rent or housing payments for 1980. All of these variables, except HE, CFR, and CFR|prime~, are available for each year and, thus, are treated as time varying covariates.

The regional amenity variables, CFR and CFR|prime~, are dummy variables. CFR comes from the response to the following question: "Suppose there were a serious emergency in your household. Is there a friend or relative living nearby whom you could call on to spend a lot of time helping out?" If the answer is "yes," CFR = 1; otherwise, CFR = 0. CFR|prime~ is similarly defined from another question about help from someone not living near the family. These variables may represent much more than the obvious factor of emergency help. The nearness of close friends and relatives may influence many aspects of household production from entertainment, to child care, to chance conversations which provide an exchange of information and a feeling of closeness. Consequently, CFR would increase A in the household production function and

thus would be expected to have a negative relationship with family migration. CFR|prime~ would increase A|prime~ and consequently would be expected to be positively related to family migration. However, since it is not clear how to empirically identify the alternative location in a two region model when there are numerous regions, the interpretation of CFR|prime~ is ambiguous. The family might have CFR|prime~ = 1 and move, but move to a location where there are no close friends or relatives. Hence, the expected relationship between CFR|prime~ and migration might be weak at best. CFR|prime~ is included in the empirical estimates for symmetry with CFR. Its exclusion does not affect the results in any appreciable manner. Both CFR and CFR|prime~ are available only for 1980 which is why they are treated as fixed covariates.

Since the housing expenditures variable, HE, is not the variable, h, in the theoretical model, further discussion of HE is needed. Since HE = hH, |Delta~R/|Delta~HE will depend upon the demand elasticity for housing. Recall that |Delta~R/|Delta~h |is less than~ 0, and hence h is positively related to migration. If the demand for housing is inelastic, then as h changes, HE will move in the same direction. Thus, both h and HE would be positively related to migration. If the demand for housing is elastic, HE will be negatively related to migration. Since housing demand is usually found to be inelastic, HE should be positively related to migration.(16)

V. The Empirical Results

The Probability of Migration

Hazard model estimates of the probability a family will migrate are shown in Table I. The estimates for two different equations are shown for each type of move. In columns (a), (c) and (e), all the variables discussed in the previous section are included in the model. In columns (b), (d) and (f), only those variables which were significant in column (a) are used. In general, the results are better for all moves, columns (a) and (b), and for moves between states, columns (e) and (f), than for moves within a state, columns (c) and (d). The results, however, are similar for the three types of moves. No statistically significant variable changes sign. Since the determinants of family migration are similar for the three types of moves, specific reference to the type of move will add little to the general discussion of the results in Table I. We will refer to column (a) when discussing the empirical results.

The signs of the coefficients are as anticipated. The weakest results are for CFR|prime~, the husband's education, |S.sub.h~, and the age variables. These variables are insignificant. The insignificance of the husband's education, the husband's age, and the wife's age may reflect problems with colinearity. In estimates not shown here, the exclusion of the wife's education, S, makes the husband's education significant. In addition, when one of the age variables was excluded, the other age variable becomes significant. CFR|prime~, however, is insignificant in all the models which were estimated. CFR|prime~ was expected to have a positive sign because the presence of close friends and relatives at alternative locations is expected to increase A|prime~ decreasing locational rent and hence increasing the probability of a move. Recall, however, that there was no way of knowing whether TABULAR DATA OMITTED a family actually moved to a location where there are close friends and relatives. Consequently, the insignificance of this variable is not surprising.

While the education variables, S and |S.sub.h~, have a positive relationship with family migration, only the wife's education is significant. Recall that the explanation for this positive sign rests on the notion that education represents general human capital. Hence, more education would increase Y|prime~/Y, for the husband's education, and would increase both w|prime~/w and A|prime~/A, for the wife's education. The significance and larger value of the estimated coefficient for the wife's education (than for the husband's education) supports the notion that there is an additional channel, A|prime~/A, through which the wife's education influences migration.

The remaining variables are both significant and have the expected signs. Since the theoretical model has clear implications for the signs of four of these variables, Y, w, CFR and HE, the results confirm the model. The husband's income, Y, has the expected negative impact on migration as does w, the wife's wage rate. Higher husband's income at the current location, ceteris paribus, makes the current location more attractive and reduces the probability the family will move. A higher wife's wage rate also increases the attractiveness of the current location and reduces family migration. Recall, that this negative impact of the wife's wage rate on migration is central to the tied-mover hypothesis. CFR has the expected negative sign. Recall that CFR was TABULAR DATA OMITTED expected to have a negative sign because the presence of close friends and relatives at the current location enhances household production, increases A, and, consequently, increases the family's locational rent reducing the probability the family will move to another region. Finally, the housing expenditures variable, HE, has a positive and significant relationship with family migration. As previously discussed, a positive relationship is expected because independent empirical studies have found an inelastic demand for housing.

The Distance of Migration

Having discussed which families are most likely to move, we will now consider whether the move was within a state or between states. Recall that Y|prime~, and w|prime~ fall with distance and that |Pi~|prime~ rises with distance because the family's market earning potential and its household technology will be adapted to its current surroundings. Letting D represent distance, |Delta~I|prime~/|Delta~D and |Delta~|Pi~|prime~/|Delta~D will depend upon the same variables which influence locational rent. Variables, like education, which are related to general human capital will make the family more adaptable to a variety of locations and thereby increase the probability that a move will be to another state. Variables, such as income and wage rates, which are location specific are likely to be negatively related to the probability that a move will be to another state.

Table II presented the logit estimates of the conditional probability that a family will move across state boundaries, given that the family moved. Of the 469 families that moved, 272 moved to another state and 197 moved to another county in their state. Estimates of the conditional probability of a move to another state are based on the 1980 value of variables defined at the original location. Only variables with a t-statistic greater than one are included in the final model. These variables are the husband's income, the wife's wage rate, housing expenditures, the wife's education and the husband's age.

In Table II, the husband's income and the wife's wage rate show a negative impact on the probability that a move was to another state. This negative impact is consistent with the notion that these variables partly reflect location specific human capital. The value of this human capital diminishes with distance. The education of the wife and the husband's age have a positive impact on the probability that a move was to another state. This positive impact is consistent with the notion that both the wife's education and the husband's age reflect general human capital. Finally, housing expenditures have a positive impact on the probability that a move will be between states. This positive impact would seem to indicate that higher priced housing is more adaptable to distant locations than is lower priced housing. This difference in adaptability might be due to less information about the quality of lower priced housing. Familiarity with the housing market diminishes with distance and this familiarity may be more important for families with lower housing expenditures.

VI. Conclusions

The theoretical model of family migration which was developed in this paper stresses the microeconomics of location choice. Notions of regional amenities and psychic costs can be given concrete interpretations within the framework of a Becker-Lancaster model of household production. Empirical hypotheses are derived from the comparative static model. An advantage of this approach is that these empirical hypotheses have straightforward interpretations within the familiar framework of choice theory. Furthermore, the empirical results support the hypotheses derived from the model. In this concluding section, we will first draw some general conclusions about the empirical results. Then we will discuss the policy implications both of the model and of the empirical results.

Both the estimates of the probability of migration and of the distance of the move are consistent with the predictions of the theoretical model. As predicted by the model, the earnings variables, which are linked to the current location, have a negative impact both on family migration and on the distance of the move. Also, as predicted by the model, variables which influence migration through their impact on general human capital, specifically the education variables, have a positive impact on family migration and on the distance of the move. That the wife's education variable was dominant over the husband's education variable suggests that there is some importance of the wife's education in household production. The age variables, although mostly reflecting indeterminacy, are positive and significant in predicting longer distanced moves. Variables related to location specific amenities, the location of close friends and relatives variables, also had the predicted signs. Finally, housing expenditures were predicted to have a positive impact on migration if the demand for housing is inelastic. Housing expenditures also had a positive impact on the distance of a move suggesting that information about the quality of less expensive housing falls with distance to a greater extent than for more expensive housing. More expensive housing is therefore easier to relocate to unfamiliar regions.

The model does have a fairly general policy implication. Families are viewed as living at their optimal location. Hence, migration is not viewed as the slow response to disequilibrium in the labor market. Consequently, increasing the rate of migration would not, per se, lead to gains in social welfare for the economy as a whole.(17) Some productivity gains could occur, through migration to regions with a higher marginal productivity of labor in producing goods. However, there would be a greater loss in the productivity of time in household production. Productivity gains could still be achieved through moving capital to regions with a lower marginal productivity of labor in producing goods. Hence, the model is consistent with the view that regional development programs are desirable.

Another policy implication of the model concerns labor market programs. Typically, job placement programs, both public and private, concentrate on locating employment for individuals. Families, however, are concerned with the employment of both husband and wife and with the household production of the family. Hence, families may be reluctant to move to unfamiliar labor markets on the basis of a job offer for the husband or wife alone. They will only move, if they believe it will be fairly easy to find suitable employment for the tied mover in the alternative location and to adjust household production to the alternative location. The positive signs of the wife's education and the husband's age in Table II support this notion. General human capital makes it more likely that a family will move to a less familiar region. A greater emphasis of employment programs on the joint employment and location decisions of the husband and wife could improve the efficiency of the labor market.

Appendix A: Finding the Shadow Price

The production functions for the two locations are

Z = A|H.sup.|Alpha~~|X.sup.|Beta~~|t.sup.1-|Alpha~-|Beta~~ (A.1)

and

Z = A|prime~|H.sup.|Alpha~~|X.sup.|Beta~~|t.sup.1-|Alpha~-|Beta~~. (A.2)

|Theta~ and L will influence household production through their impact on A and A|prime~. When t |is less than~ T, the wife is employed and the budget constraint for commodities at each location can be written as

I = Y + wT = |Pi~Z, (A.3)

for the current location, and as

I|prime~ = Y|prime~ + w|prime~T = |Pi~|prime~Z, (A.4)

for the alternative location.

To find the shadow price, |Pi~, in (A.3) note that the budget for goods can be written as

I = Y + wT = hH + pX + wt. (A.5)

The first step in finding |Pi~ is to solve for H and X as functions of t. Optimal production of commodities implies that the household will use inputs up to the point where

(|Delta~Z/|Delta~X)/p = (|Delta~Z/|Delta~t)/w

and

(|Delta~Z/|Delta~H)/h = (|Delta~Z/|Delta~t)/w. (A.6)

By differentiating (A.1), simplifying, and substituting into (A.6), X and H can be found as functions of t yielding,

X = (w/p)(|Beta~/(1 - |Alpha~ - |Beta~))t

and

H = (w/h)(|Alpha~/(1 - |Alpha~ - |Beta~))t. (A.7)

Note that (A.7) implies that X/t and H/t have unique solutions determined by the production elasticities and by w, p, and h.

The second step in finding |Pi~ is to express t as a function of Z. Taking advantage of linear homogeneity and dividing (A.1) by t yields

Z/t = A||H/t~.sup.|Alpha~~||X/t~.sup.|Beta~~. (A.8)

Then substituting X/t and H/t, from (A.7), into (A.8), yields

Z/t = A||(w/h)(|Alpha~/(1 - |Alpha~ - |Beta~))~.sup.|Alpha~~||(w/p)(|Beta~/(1 - |Alpha~ - |Beta~))~.sup.|Beta~~. (A.9)

We can now find |Pi~ by solving (A.9) for t, substituting for t in (A.7) to find H, X, and t as functions of Z. Substituting these expressions for H, X and t into the budget constraint (A.5) and simplifying yields |Pi~Z, where

|Pi~ = w/A(1 - |Alpha~ - |Beta~)||(w/h)(|Alpha~/(1 - |Alpha~ - |Beta~))~.sup.|Alpha~~||(w/p)(|Beta~/(1 - |Alpha~ - |Beta~))~.sup.|Beta~~. (A.10)

Similarly, for the alternative location,

|Pi~|prime~ = w|prime~/A|prime~(1 - |Alpha~ - |Beta~)||(w|prime~/h|prime~)(|Alpha~/(1 - |Alpha~ - |Beta~))~.sup.|Alpha~~||(w|prime~/p|prime~)(|Beta~/(1 - |Alpha~ - |Beta~))~.sup.|Beta~~. (A.11)

Hence, the ratio of shadow prices |Pi~|prime~/|Pi~ can be found by dividing (A.10) into (A.11) and simplifying, yielding

|Pi~|prime~/|Pi~ = (A/A|prime~)|(h|prime~/h).sup.|Alpha~~|(p|prime~/p).sup.|Beta~~ |(w|prime~/w).sup.1-|Alpha~-|Beta~~. (A.12)

When the wife specializes in household production, t = T, the commodity budget constraints are nonlinear and become

I = Y = q|Z.sup.1/(|Alpha~+|Beta~)~ and

I|prime~ = Y|prime~ = q|prime~|Z.sup.1/(|Alpha~+|Beta~)~. (A.13)

The same approach as before yields

q = {p(|Alpha~ + |Beta~)/|Beta~}/{|A.sup.1/(|Alpha~+|Beta~)~|T.sup.(1-|Alpha~-|B eta~)/(|Alpha~+|Beta~)~||(p/h)(|Alpha~/|Beta~)~.sup.|Alpha~/(|A lpha~+|Beta~)~}, (A.14)

and

q|prime~ = {p|prime~(|Alpha~ + |Beta~)/|Beta~}/{|A|prime~.sup.1/|Alpha~+|Beta~)~|T.sup.(1-|Alp ha~-|Beta~)/(|Alpha~+|Beta~)~||(p|prime~/h|prime~)(|Alpha~/|Bet a~)~.sup.|Alpha~/(|Alpha~+|Beta~)~}. (A.15)

Dividing (A.14) into (A.15) yields, upon simplification,

q|prime~/q = ||(A/A|prime~)|(h|prime~/h).sup.|Alpha~~|(p|prime~/p).sup.|Beta ~~~.sup.1/(|Alpha~+|Beta~)~. (A.16)

Appendix B: Education and Locational Rent

To see the impact of the husband's education, |S.sub.h~, on locational rent, first note that

|Delta~(Y|prime~/Y)/|Delta~|S.sub.h~ = (|Delta~Y|prime~/|Delta~|S.sub.h~)/Y - (Y|prime~/|Y.sup.2~)(|Delta~Y/|Delta~|S.sub.h~). (B.1)

Next, differentiate R with respect to |S.sub.h~ yielding

|Delta~R/|Delta~|S.sub.h~ = (|Delta~Y/|Delta~|S.sub.h~)(|Pi~|prime~/|Pi~) - |Delta~Y|prime~/|Delta~|S.sub.h~. (B.2)

Rearranging terms in (B.1) and solving for |Delta~Y|prime~/|Delta~|S.sub.h~ and then substituting into (B.2) yields

|Delta~R/|Delta~|S.sub.h~ = |(|Delta~Y/|Delta~|S.sub.h~)(|Pi~|prime~/|Pi~ - Y|prime~/Y)~ - |Y(|Delta~(Y|prime~/Y)/|Delta~|S.sub.h~)~. (B.3)

To interpret equation (B.3), note that since |Delta~Y/|Delta~|S.sub.h~ |is greater than~ 0, the term in the first brackets will be positive if

(|Pi~|prime~/|Pi~ - Y|prime~/Y) |is greater than~ 0. (B.4)

Recall that, since the current location is, by assumption, the optimum location, R |is greater than~ 0. Hence, rearranging terms in equation (3) yields

(|Pi~|prime~/|Pi~ - I|prime~/I) |is greater than~ 0. (B.5)

Since husband's income is a component of full income, we would expect (B.4) to usually hold when (B.5) holds making the sign of the first bracketed term often positive. The usual argument that education represents general human capital will make |Delta~(Y|prime~/Y)/|Delta~|S.sub.h~ positive and hence make the second bracketed term positive. A positive second bracketed term could result in the husband's education having a negative impact on locational rent and, hence, a positive impact on migration.

A similar argument can be applied to the wife's education, S. The impact of the wife's education on locational rent is

|Delta~R/|Delta~S = |(|Delta~w/|Delta~S)T(|Pi~|prime~/|Pi~ - w|prime~/w~

- |(wT(|Delta~Y|prime~/Y)/|Delta~S)(1 - (1 - |Alpha~ - |Beta~)|Pi~|prime~/|Pi~(w|prime~/w))~

- ||Delta~(A/A|prime~)/|Delta~S)(IA|prime~|Pi~|prime~/A|Pi~)~. (B.6)

The term in the third brackets reflects the impact of the wife's education on household production. Uncertainty as to the sign of this term complicates the discussion. However, the same argument as with the husband's education applies to the first bracketed expression. The second bracketed expression is also likely to be positive. It will be positive unless

|Pi~|prime~/|Pi~ - (w|prime~/w)(1/(1 - |Alpha~ - |Beta~)) |is greater than~ 0. (B.7)

Since 1/(1 - |Alpha~ - |Beta~) |is greater than~ 1, inequality (B.7) would imply considerable regional differences in the costs of household production. In this situation, the wife's education could be positively associated with locational rent. Otherwise, |Delta~R/|Delta~S is likely to be negative.

Much of the theoretical part of this paper was written while on sabbatical at the University of Essex. We wish to thank the anonymous referee for useful suggestions about both the content and the organization of the paper.

1. Others looking at the wife's income and migration include Long |22~, Mincer and Polachek |27~, DaVanzo |7~, Polachek and Horvath |28~, Sandell |33~, and Bartel |1~.

2. While household production models have been applied to many areas of household choice including fertility, health care, marriage and divorce, these models have only recently received attention in the migration literature. See Shields and Shields |39~ for an analytical survey of the migration literature which includes a section on household production and migration.

3. Housing costs play an important role in hedonic wage models which stress that high wage rates or low housing costs are compensation for poor regional amenities. See Rosen |32~, Roback |30; 31~, Graves |10~, Blomquist, Berger and Hoehn |3~ and Knapp and Graves |17~.

4. In this model, the wife's time plays a dual role. It can be used in household production or in earning income. Note that the husband's time does not appear in the household production function. Thus, income can be divided into a component which is independent of the time used in household production, called the husband's income (Y), and a component, called the wife's earnings, w(T - t), which is partly determined by the time used in household production, t. While this treatment of time is not crucial for the migration model which will be developed, it is made because we wish to emphasize the labor market decisions of the tied mover.

5. If the wife completely specializes in household production, the implicit constraint for commodities is nonlinear. See Willis |41~ and Pollak and Wachter |29~.

6. For a discussion, see Greenwood |12~.

7. For a similar argument, expressed in terms of labor market search and the reservation wage rate, see Denslow and Eaton |8~. They express their argument in terms of the reservation wage rate at the alternative location for moving to that location.

8. Since there is only one aggregate commodity, U(Z) is maximized when Z is maximized if |Delta~U/|Delta~Z |is greater than~ 0 for all Z.

9. For a consideration of locations being chosen because they favor the production of certain commodities, see T. P. Schultz |34~.

10. A positive relationship between education and individual migration has usually been found. See Greenwood |11~, Bowles |4~, Long |21~, and Fields |9~. Explanations have included the idea that education will reduce the risks of a move either because education creates more employment opportunities |37~ or because education contributes to an individual's ability to collect and process information |35; 36~.

11. The maximum age of the husband for any year of this study is fifty-five. See Mincer |25~ for a discussion of the affects of both age and education on human capital.

12. See Bowles |4~ and Schwartz |37~ for a discussion of age and individual migration.

13. See Levy and Wadycki |20~, Schwartz |36~, Denslow and Eaton |8~, and Clark and Cosgrove |5~.

14. The hazard model was estimated using BMDP |15~. For a discussion of the hazard model see Cox |6~, Heckman and Singer |13~, and Tony Lancaster |19~. For the logit model, see Madalla |23~.

15. The exclusion of these variables will lead to biased estimates if they are correlated with variables in the estimated model. For example, Y and Y|prime~ would be correlated. However, since they have opposite effects on migration, the bias would be towards the null hypothesis.

16. See Houthakker and Taylor |16~ and Mansur and Whalley |24~ for studies which found an inelastic demand for housing.

17. See Herzog and Schlottman |14~ for a treatment of migration, allocative efficiency and the role of psychic versus informational and physical moving costs.

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Author: | Shields, Gail M. |
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Publication: | Southern Economic Journal |

Date: | Apr 1, 1993 |

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